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Ben Green and I have just uploaded our joint paper, “The distribution of polynomials over finite fields, with applications to the Gowers norms“, to the arXiv, and submitted to Contributions to Discrete Mathematics. This paper, which we first announced at the recent FOCS meeting, and then gave an update on two weeks ago on this blog, is now in final form. It is being made available simultaneously with a closely related paper of Lovett, Meshulam, and Samorodnitsky.

In the previous post on this topic, I focused on the negative results in the paper, and in particular the fact that the inverse conjecture for the Gowers norm fails for certain degrees in low characteristic. Today, I’d like to focus instead on the positive results, which assert that for polynomials in many variables over finite fields whose degree is less than the characteristic of the field, one has a satisfactory theory for the distribution of these polynomials. Very roughly speaking, the main technical results are:

• A regularity lemma: Any polynomial can be expressed as a combination of a bounded number of other polynomials which are regular, in the sense that no non-trivial linear combination of these polynomials can be expressed efficiently in terms of lower degree polynomials.
• A counting lemma: A regular collection of polynomials behaves as if the polynomials were selected randomly. In particular, the polynomials are jointly equidistributed.

Recently, I had tentatively announced a forthcoming result with Ben Green establishing the “Gowers inverse conjecture” (or more accurately, the “inverse conjecture for the Gowers uniformity norm”) for vector spaces ${\Bbb F}_p^n$ over a finite field ${\Bbb F}_p$, in the special case when p=2 and when the function $f: {\Bbb F}_p^n \to {\Bbb C}$ for which the inverse conjecture is to be applied is assumed to be a polynomial phase of bounded degree (thus $f= e^{2\pi i P/|{\Bbb F}|}$, where $P: {\Bbb F}_p^n \to {\Bbb F}_p$ is a polynomial of some degree $d=O(1)$). See my FOCS article for some further discussion of this conjecture, which has applications to both polynomiality testing and to various structural decompositions involving the Gowers norm.

This conjecture can be informally stated as follows. By iterating the obvious fact that the derivative of a polynomial of degree at most d is a polynomial of degree at most d-1, we see that a function $P: {\Bbb F}_p^n \to {\Bbb F}_p$ is a polynomial of degree at most d if and only if

$\sum_{\omega_1,\ldots,\omega_{d+1} \in \{0,1\}} (-1)^{\omega_1+\ldots+\omega_{d+1}} P(x +\omega_1 h_1 + \ldots + \omega_{d+1} h_{d+1}) = 0$

for all $x,h_1,\ldots,h_{d+1} \in {\Bbb F}_p^n$. From this one can deduce that a function $f: {\Bbb F}_p^n \to {\Bbb C}$ bounded in magnitude by 1 is a polynomial phase of degree at most d if and only if the Gowers norm

$\|f\|_{U^{d+1}({\Bbb F}_p^n)} := \bigl( {\Bbb E}_{x,h_1,\ldots,h_{d+1} \in {\Bbb F}_p^n} \prod_{\omega_1,\ldots,\omega_{d+1} \in \{0,1\}}$

${\mathcal C}^{\omega_1+\ldots+\omega_{d+1}} f(x + \omega_1 h_1 + \ldots + \omega_{d+1} h_{d+1}) \bigr)^{1/2^{d+1}}$

is equal to its maximal value of 1. The inverse conjecture for the Gowers norm, in its usual formulation, says that, more generally, if a function $f: {\Bbb F}_p^n \to {\Bbb C}$ bounded in magnitude by 1 has large Gowers norm (e.g. $\|f\|_{U^{d+1}} \geq \varepsilon$) then f has some non-trivial correlation with some polynomial phase g (e.g. $\langle f, g \rangle > c(\varepsilon)$ for some $c(\varepsilon) > 0$). Informally, this conjecture asserts that if a function has biased $(d+1)^{th}$ derivatives, then one should be able to “integrate” this bias and conclude that the function is biased relative to a polynomial of degree d. The conjecture has already been proven for $d \leq 2$. There are analogues of this conjecture for cyclic groups which are of relevance to Szemerédi’s theorem and to counting linear patterns in primes, but I will not discuss those here.

At the time of the announcement, our paper had not quite been fully written up. This turned out to be a little unfortunate, because soon afterwards we discovered that our arguments at one point had to go through a version of Newton’s interpolation formula, which involves a factor of d! in the denominator and so is only valid when the characteristic p of the field exceeds the degree. So our arguments in fact are only valid in the range $p > d$, and in particular are rather trivial in the important case $p=2$; my previous announcement should thus be amended accordingly.